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ai-content-maker/.venv/Lib/site-packages/sympy/concrete/tests/test_delta.py

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2024-05-03 04:18:51 +03:00
from sympy.concrete import Sum
from sympy.concrete.delta import deltaproduct as dp, deltasummation as ds, _extract_delta
from sympy.core import Eq, S, symbols, oo
from sympy.functions import KroneckerDelta as KD, Piecewise, piecewise_fold
from sympy.logic import And
from sympy.testing.pytest import raises
i, j, k, l, m = symbols("i j k l m", integer=True, finite=True)
x, y = symbols("x y", commutative=False)
def test_deltaproduct_trivial():
assert dp(x, (j, 1, 0)) == 1
assert dp(x, (j, 1, 3)) == x**3
assert dp(x + y, (j, 1, 3)) == (x + y)**3
assert dp(x*y, (j, 1, 3)) == (x*y)**3
assert dp(KD(i, j), (k, 1, 3)) == KD(i, j)
assert dp(x*KD(i, j), (k, 1, 3)) == x**3*KD(i, j)
assert dp(x*y*KD(i, j), (k, 1, 3)) == (x*y)**3*KD(i, j)
def test_deltaproduct_basic():
assert dp(KD(i, j), (j, 1, 3)) == 0
assert dp(KD(i, j), (j, 1, 1)) == KD(i, 1)
assert dp(KD(i, j), (j, 2, 2)) == KD(i, 2)
assert dp(KD(i, j), (j, 3, 3)) == KD(i, 3)
assert dp(KD(i, j), (j, 1, k)) == KD(i, 1)*KD(k, 1) + KD(k, 0)
assert dp(KD(i, j), (j, k, 3)) == KD(i, 3)*KD(k, 3) + KD(k, 4)
assert dp(KD(i, j), (j, k, l)) == KD(i, l)*KD(k, l) + KD(k, l + 1)
def test_deltaproduct_mul_x_kd():
assert dp(x*KD(i, j), (j, 1, 3)) == 0
assert dp(x*KD(i, j), (j, 1, 1)) == x*KD(i, 1)
assert dp(x*KD(i, j), (j, 2, 2)) == x*KD(i, 2)
assert dp(x*KD(i, j), (j, 3, 3)) == x*KD(i, 3)
assert dp(x*KD(i, j), (j, 1, k)) == x*KD(i, 1)*KD(k, 1) + KD(k, 0)
assert dp(x*KD(i, j), (j, k, 3)) == x*KD(i, 3)*KD(k, 3) + KD(k, 4)
assert dp(x*KD(i, j), (j, k, l)) == x*KD(i, l)*KD(k, l) + KD(k, l + 1)
def test_deltaproduct_mul_add_x_y_kd():
assert dp((x + y)*KD(i, j), (j, 1, 3)) == 0
assert dp((x + y)*KD(i, j), (j, 1, 1)) == (x + y)*KD(i, 1)
assert dp((x + y)*KD(i, j), (j, 2, 2)) == (x + y)*KD(i, 2)
assert dp((x + y)*KD(i, j), (j, 3, 3)) == (x + y)*KD(i, 3)
assert dp((x + y)*KD(i, j), (j, 1, k)) == \
(x + y)*KD(i, 1)*KD(k, 1) + KD(k, 0)
assert dp((x + y)*KD(i, j), (j, k, 3)) == \
(x + y)*KD(i, 3)*KD(k, 3) + KD(k, 4)
assert dp((x + y)*KD(i, j), (j, k, l)) == \
(x + y)*KD(i, l)*KD(k, l) + KD(k, l + 1)
def test_deltaproduct_add_kd_kd():
assert dp(KD(i, k) + KD(j, k), (k, 1, 3)) == 0
assert dp(KD(i, k) + KD(j, k), (k, 1, 1)) == KD(i, 1) + KD(j, 1)
assert dp(KD(i, k) + KD(j, k), (k, 2, 2)) == KD(i, 2) + KD(j, 2)
assert dp(KD(i, k) + KD(j, k), (k, 3, 3)) == KD(i, 3) + KD(j, 3)
assert dp(KD(i, k) + KD(j, k), (k, 1, l)) == KD(l, 0) + \
KD(i, 1)*KD(l, 1) + KD(j, 1)*KD(l, 1) + \
KD(i, 1)*KD(j, 2)*KD(l, 2) + KD(j, 1)*KD(i, 2)*KD(l, 2)
assert dp(KD(i, k) + KD(j, k), (k, l, 3)) == KD(l, 4) + \
KD(i, 3)*KD(l, 3) + KD(j, 3)*KD(l, 3) + \
KD(i, 2)*KD(j, 3)*KD(l, 2) + KD(i, 3)*KD(j, 2)*KD(l, 2)
assert dp(KD(i, k) + KD(j, k), (k, l, m)) == KD(l, m + 1) + \
KD(i, m)*KD(l, m) + KD(j, m)*KD(l, m) + \
KD(i, m)*KD(j, m - 1)*KD(l, m - 1) + KD(i, m - 1)*KD(j, m)*KD(l, m - 1)
def test_deltaproduct_mul_x_add_kd_kd():
assert dp(x*(KD(i, k) + KD(j, k)), (k, 1, 3)) == 0
assert dp(x*(KD(i, k) + KD(j, k)), (k, 1, 1)) == x*(KD(i, 1) + KD(j, 1))
assert dp(x*(KD(i, k) + KD(j, k)), (k, 2, 2)) == x*(KD(i, 2) + KD(j, 2))
assert dp(x*(KD(i, k) + KD(j, k)), (k, 3, 3)) == x*(KD(i, 3) + KD(j, 3))
assert dp(x*(KD(i, k) + KD(j, k)), (k, 1, l)) == KD(l, 0) + \
x*KD(i, 1)*KD(l, 1) + x*KD(j, 1)*KD(l, 1) + \
x**2*KD(i, 1)*KD(j, 2)*KD(l, 2) + x**2*KD(j, 1)*KD(i, 2)*KD(l, 2)
assert dp(x*(KD(i, k) + KD(j, k)), (k, l, 3)) == KD(l, 4) + \
x*KD(i, 3)*KD(l, 3) + x*KD(j, 3)*KD(l, 3) + \
x**2*KD(i, 2)*KD(j, 3)*KD(l, 2) + x**2*KD(i, 3)*KD(j, 2)*KD(l, 2)
assert dp(x*(KD(i, k) + KD(j, k)), (k, l, m)) == KD(l, m + 1) + \
x*KD(i, m)*KD(l, m) + x*KD(j, m)*KD(l, m) + \
x**2*KD(i, m - 1)*KD(j, m)*KD(l, m - 1) + \
x**2*KD(i, m)*KD(j, m - 1)*KD(l, m - 1)
def test_deltaproduct_mul_add_x_y_add_kd_kd():
assert dp((x + y)*(KD(i, k) + KD(j, k)), (k, 1, 3)) == 0
assert dp((x + y)*(KD(i, k) + KD(j, k)), (k, 1, 1)) == \
(x + y)*(KD(i, 1) + KD(j, 1))
assert dp((x + y)*(KD(i, k) + KD(j, k)), (k, 2, 2)) == \
(x + y)*(KD(i, 2) + KD(j, 2))
assert dp((x + y)*(KD(i, k) + KD(j, k)), (k, 3, 3)) == \
(x + y)*(KD(i, 3) + KD(j, 3))
assert dp((x + y)*(KD(i, k) + KD(j, k)), (k, 1, l)) == KD(l, 0) + \
(x + y)*KD(i, 1)*KD(l, 1) + (x + y)*KD(j, 1)*KD(l, 1) + \
(x + y)**2*KD(i, 1)*KD(j, 2)*KD(l, 2) + \
(x + y)**2*KD(j, 1)*KD(i, 2)*KD(l, 2)
assert dp((x + y)*(KD(i, k) + KD(j, k)), (k, l, 3)) == KD(l, 4) + \
(x + y)*KD(i, 3)*KD(l, 3) + (x + y)*KD(j, 3)*KD(l, 3) + \
(x + y)**2*KD(i, 2)*KD(j, 3)*KD(l, 2) + \
(x + y)**2*KD(i, 3)*KD(j, 2)*KD(l, 2)
assert dp((x + y)*(KD(i, k) + KD(j, k)), (k, l, m)) == KD(l, m + 1) + \
(x + y)*KD(i, m)*KD(l, m) + (x + y)*KD(j, m)*KD(l, m) + \
(x + y)**2*KD(i, m - 1)*KD(j, m)*KD(l, m - 1) + \
(x + y)**2*KD(i, m)*KD(j, m - 1)*KD(l, m - 1)
def test_deltaproduct_add_mul_x_y_mul_x_kd():
assert dp(x*y + x*KD(i, j), (j, 1, 3)) == (x*y)**3 + \
x*(x*y)**2*KD(i, 1) + (x*y)*x*(x*y)*KD(i, 2) + (x*y)**2*x*KD(i, 3)
assert dp(x*y + x*KD(i, j), (j, 1, 1)) == x*y + x*KD(i, 1)
assert dp(x*y + x*KD(i, j), (j, 2, 2)) == x*y + x*KD(i, 2)
assert dp(x*y + x*KD(i, j), (j, 3, 3)) == x*y + x*KD(i, 3)
assert dp(x*y + x*KD(i, j), (j, 1, k)) == \
(x*y)**k + Piecewise(
((x*y)**(i - 1)*x*(x*y)**(k - i), And(1 <= i, i <= k)),
(0, True)
)
assert dp(x*y + x*KD(i, j), (j, k, 3)) == \
(x*y)**(-k + 4) + Piecewise(
((x*y)**(i - k)*x*(x*y)**(3 - i), And(k <= i, i <= 3)),
(0, True)
)
assert dp(x*y + x*KD(i, j), (j, k, l)) == \
(x*y)**(-k + l + 1) + Piecewise(
((x*y)**(i - k)*x*(x*y)**(l - i), And(k <= i, i <= l)),
(0, True)
)
def test_deltaproduct_mul_x_add_y_kd():
assert dp(x*(y + KD(i, j)), (j, 1, 3)) == (x*y)**3 + \
x*(x*y)**2*KD(i, 1) + (x*y)*x*(x*y)*KD(i, 2) + (x*y)**2*x*KD(i, 3)
assert dp(x*(y + KD(i, j)), (j, 1, 1)) == x*(y + KD(i, 1))
assert dp(x*(y + KD(i, j)), (j, 2, 2)) == x*(y + KD(i, 2))
assert dp(x*(y + KD(i, j)), (j, 3, 3)) == x*(y + KD(i, 3))
assert dp(x*(y + KD(i, j)), (j, 1, k)) == \
(x*y)**k + Piecewise(
((x*y)**(i - 1)*x*(x*y)**(k - i), And(1 <= i, i <= k)),
(0, True)
).expand()
assert dp(x*(y + KD(i, j)), (j, k, 3)) == \
((x*y)**(-k + 4) + Piecewise(
((x*y)**(i - k)*x*(x*y)**(3 - i), And(k <= i, i <= 3)),
(0, True)
)).expand()
assert dp(x*(y + KD(i, j)), (j, k, l)) == \
((x*y)**(-k + l + 1) + Piecewise(
((x*y)**(i - k)*x*(x*y)**(l - i), And(k <= i, i <= l)),
(0, True)
)).expand()
def test_deltaproduct_mul_x_add_y_twokd():
assert dp(x*(y + 2*KD(i, j)), (j, 1, 3)) == (x*y)**3 + \
2*x*(x*y)**2*KD(i, 1) + 2*x*y*x*x*y*KD(i, 2) + 2*(x*y)**2*x*KD(i, 3)
assert dp(x*(y + 2*KD(i, j)), (j, 1, 1)) == x*(y + 2*KD(i, 1))
assert dp(x*(y + 2*KD(i, j)), (j, 2, 2)) == x*(y + 2*KD(i, 2))
assert dp(x*(y + 2*KD(i, j)), (j, 3, 3)) == x*(y + 2*KD(i, 3))
assert dp(x*(y + 2*KD(i, j)), (j, 1, k)) == \
(x*y)**k + Piecewise(
(2*(x*y)**(i - 1)*x*(x*y)**(k - i), And(1 <= i, i <= k)),
(0, True)
).expand()
assert dp(x*(y + 2*KD(i, j)), (j, k, 3)) == \
((x*y)**(-k + 4) + Piecewise(
(2*(x*y)**(i - k)*x*(x*y)**(3 - i), And(k <= i, i <= 3)),
(0, True)
)).expand()
assert dp(x*(y + 2*KD(i, j)), (j, k, l)) == \
((x*y)**(-k + l + 1) + Piecewise(
(2*(x*y)**(i - k)*x*(x*y)**(l - i), And(k <= i, i <= l)),
(0, True)
)).expand()
def test_deltaproduct_mul_add_x_y_add_y_kd():
assert dp((x + y)*(y + KD(i, j)), (j, 1, 3)) == ((x + y)*y)**3 + \
(x + y)*((x + y)*y)**2*KD(i, 1) + \
(x + y)*y*(x + y)**2*y*KD(i, 2) + \
((x + y)*y)**2*(x + y)*KD(i, 3)
assert dp((x + y)*(y + KD(i, j)), (j, 1, 1)) == (x + y)*(y + KD(i, 1))
assert dp((x + y)*(y + KD(i, j)), (j, 2, 2)) == (x + y)*(y + KD(i, 2))
assert dp((x + y)*(y + KD(i, j)), (j, 3, 3)) == (x + y)*(y + KD(i, 3))
assert dp((x + y)*(y + KD(i, j)), (j, 1, k)) == \
((x + y)*y)**k + Piecewise(
(((x + y)*y)**(-1)*((x + y)*y)**i*(x + y)*((x + y)*y
)**k*((x + y)*y)**(-i), (i >= 1) & (i <= k)), (0, True))
assert dp((x + y)*(y + KD(i, j)), (j, k, 3)) == (
(x + y)*y)**4*((x + y)*y)**(-k) + Piecewise((((x + y)*y)**i*(
(x + y)*y)**(-k)*(x + y)*((x + y)*y)**3*((x + y)*y)**(-i),
(i >= k) & (i <= 3)), (0, True))
assert dp((x + y)*(y + KD(i, j)), (j, k, l)) == \
(x + y)*y*((x + y)*y)**l*((x + y)*y)**(-k) + Piecewise(
(((x + y)*y)**i*((x + y)*y)**(-k)*(x + y)*((x + y)*y
)**l*((x + y)*y)**(-i), (i >= k) & (i <= l)), (0, True))
def test_deltaproduct_mul_add_x_kd_add_y_kd():
assert dp((x + KD(i, k))*(y + KD(i, j)), (j, 1, 3)) == \
KD(i, 1)*(KD(i, k) + x)*((KD(i, k) + x)*y)**2 + \
KD(i, 2)*(KD(i, k) + x)*y*(KD(i, k) + x)**2*y + \
KD(i, 3)*((KD(i, k) + x)*y)**2*(KD(i, k) + x) + \
((KD(i, k) + x)*y)**3
assert dp((x + KD(i, k))*(y + KD(i, j)), (j, 1, 1)) == \
(x + KD(i, k))*(y + KD(i, 1))
assert dp((x + KD(i, k))*(y + KD(i, j)), (j, 2, 2)) == \
(x + KD(i, k))*(y + KD(i, 2))
assert dp((x + KD(i, k))*(y + KD(i, j)), (j, 3, 3)) == \
(x + KD(i, k))*(y + KD(i, 3))
assert dp((x + KD(i, k))*(y + KD(i, j)), (j, 1, k)) == \
((KD(i, k) + x)*y)**k + Piecewise(
(((KD(i, k) + x)*y)**(-1)*((KD(i, k) + x)*y)**i*(KD(i, k) + x
)*((KD(i, k) + x)*y)**k*((KD(i, k) + x)*y)**(-i), (i >= 1
) & (i <= k)), (0, True))
assert dp((x + KD(i, k))*(y + KD(i, j)), (j, k, 3)) == (
(KD(i, k) + x)*y)**4*((KD(i, k) + x)*y)**(-k) + Piecewise(
(((KD(i, k) + x)*y)**i*((KD(i, k) + x)*y)**(-k)*(KD(i, k)
+ x)*((KD(i, k) + x)*y)**3*((KD(i, k) + x)*y)**(-i),
(i >= k) & (i <= 3)), (0, True))
assert dp((x + KD(i, k))*(y + KD(i, j)), (j, k, l)) == (
KD(i, k) + x)*y*((KD(i, k) + x)*y)**l*((KD(i, k) + x)*y
)**(-k) + Piecewise((((KD(i, k) + x)*y)**i*((KD(i, k) + x
)*y)**(-k)*(KD(i, k) + x)*((KD(i, k) + x)*y)**l*((KD(i, k) + x
)*y)**(-i), (i >= k) & (i <= l)), (0, True))
def test_deltasummation_trivial():
assert ds(x, (j, 1, 0)) == 0
assert ds(x, (j, 1, 3)) == 3*x
assert ds(x + y, (j, 1, 3)) == 3*(x + y)
assert ds(x*y, (j, 1, 3)) == 3*x*y
assert ds(KD(i, j), (k, 1, 3)) == 3*KD(i, j)
assert ds(x*KD(i, j), (k, 1, 3)) == 3*x*KD(i, j)
assert ds(x*y*KD(i, j), (k, 1, 3)) == 3*x*y*KD(i, j)
def test_deltasummation_basic_numerical():
n = symbols('n', integer=True, nonzero=True)
assert ds(KD(n, 0), (n, 1, 3)) == 0
# return unevaluated, until it gets implemented
assert ds(KD(i**2, j**2), (j, -oo, oo)) == \
Sum(KD(i**2, j**2), (j, -oo, oo))
assert Piecewise((KD(i, k), And(1 <= i, i <= 3)), (0, True)) == \
ds(KD(i, j)*KD(j, k), (j, 1, 3)) == \
ds(KD(j, k)*KD(i, j), (j, 1, 3))
assert ds(KD(i, k), (k, -oo, oo)) == 1
assert ds(KD(i, k), (k, 0, oo)) == Piecewise((1, S.Zero <= i), (0, True))
assert ds(KD(i, k), (k, 1, 3)) == \
Piecewise((1, And(1 <= i, i <= 3)), (0, True))
assert ds(k*KD(i, j)*KD(j, k), (k, -oo, oo)) == j*KD(i, j)
assert ds(j*KD(i, j), (j, -oo, oo)) == i
assert ds(i*KD(i, j), (i, -oo, oo)) == j
assert ds(x, (i, 1, 3)) == 3*x
assert ds((i + j)*KD(i, j), (j, -oo, oo)) == 2*i
def test_deltasummation_basic_symbolic():
assert ds(KD(i, j), (j, 1, 3)) == \
Piecewise((1, And(1 <= i, i <= 3)), (0, True))
assert ds(KD(i, j), (j, 1, 1)) == Piecewise((1, Eq(i, 1)), (0, True))
assert ds(KD(i, j), (j, 2, 2)) == Piecewise((1, Eq(i, 2)), (0, True))
assert ds(KD(i, j), (j, 3, 3)) == Piecewise((1, Eq(i, 3)), (0, True))
assert ds(KD(i, j), (j, 1, k)) == \
Piecewise((1, And(1 <= i, i <= k)), (0, True))
assert ds(KD(i, j), (j, k, 3)) == \
Piecewise((1, And(k <= i, i <= 3)), (0, True))
assert ds(KD(i, j), (j, k, l)) == \
Piecewise((1, And(k <= i, i <= l)), (0, True))
def test_deltasummation_mul_x_kd():
assert ds(x*KD(i, j), (j, 1, 3)) == \
Piecewise((x, And(1 <= i, i <= 3)), (0, True))
assert ds(x*KD(i, j), (j, 1, 1)) == Piecewise((x, Eq(i, 1)), (0, True))
assert ds(x*KD(i, j), (j, 2, 2)) == Piecewise((x, Eq(i, 2)), (0, True))
assert ds(x*KD(i, j), (j, 3, 3)) == Piecewise((x, Eq(i, 3)), (0, True))
assert ds(x*KD(i, j), (j, 1, k)) == \
Piecewise((x, And(1 <= i, i <= k)), (0, True))
assert ds(x*KD(i, j), (j, k, 3)) == \
Piecewise((x, And(k <= i, i <= 3)), (0, True))
assert ds(x*KD(i, j), (j, k, l)) == \
Piecewise((x, And(k <= i, i <= l)), (0, True))
def test_deltasummation_mul_add_x_y_kd():
assert ds((x + y)*KD(i, j), (j, 1, 3)) == \
Piecewise((x + y, And(1 <= i, i <= 3)), (0, True))
assert ds((x + y)*KD(i, j), (j, 1, 1)) == \
Piecewise((x + y, Eq(i, 1)), (0, True))
assert ds((x + y)*KD(i, j), (j, 2, 2)) == \
Piecewise((x + y, Eq(i, 2)), (0, True))
assert ds((x + y)*KD(i, j), (j, 3, 3)) == \
Piecewise((x + y, Eq(i, 3)), (0, True))
assert ds((x + y)*KD(i, j), (j, 1, k)) == \
Piecewise((x + y, And(1 <= i, i <= k)), (0, True))
assert ds((x + y)*KD(i, j), (j, k, 3)) == \
Piecewise((x + y, And(k <= i, i <= 3)), (0, True))
assert ds((x + y)*KD(i, j), (j, k, l)) == \
Piecewise((x + y, And(k <= i, i <= l)), (0, True))
def test_deltasummation_add_kd_kd():
assert ds(KD(i, k) + KD(j, k), (k, 1, 3)) == piecewise_fold(
Piecewise((1, And(1 <= i, i <= 3)), (0, True)) +
Piecewise((1, And(1 <= j, j <= 3)), (0, True)))
assert ds(KD(i, k) + KD(j, k), (k, 1, 1)) == piecewise_fold(
Piecewise((1, Eq(i, 1)), (0, True)) +
Piecewise((1, Eq(j, 1)), (0, True)))
assert ds(KD(i, k) + KD(j, k), (k, 2, 2)) == piecewise_fold(
Piecewise((1, Eq(i, 2)), (0, True)) +
Piecewise((1, Eq(j, 2)), (0, True)))
assert ds(KD(i, k) + KD(j, k), (k, 3, 3)) == piecewise_fold(
Piecewise((1, Eq(i, 3)), (0, True)) +
Piecewise((1, Eq(j, 3)), (0, True)))
assert ds(KD(i, k) + KD(j, k), (k, 1, l)) == piecewise_fold(
Piecewise((1, And(1 <= i, i <= l)), (0, True)) +
Piecewise((1, And(1 <= j, j <= l)), (0, True)))
assert ds(KD(i, k) + KD(j, k), (k, l, 3)) == piecewise_fold(
Piecewise((1, And(l <= i, i <= 3)), (0, True)) +
Piecewise((1, And(l <= j, j <= 3)), (0, True)))
assert ds(KD(i, k) + KD(j, k), (k, l, m)) == piecewise_fold(
Piecewise((1, And(l <= i, i <= m)), (0, True)) +
Piecewise((1, And(l <= j, j <= m)), (0, True)))
def test_deltasummation_add_mul_x_kd_kd():
assert ds(x*KD(i, k) + KD(j, k), (k, 1, 3)) == piecewise_fold(
Piecewise((x, And(1 <= i, i <= 3)), (0, True)) +
Piecewise((1, And(1 <= j, j <= 3)), (0, True)))
assert ds(x*KD(i, k) + KD(j, k), (k, 1, 1)) == piecewise_fold(
Piecewise((x, Eq(i, 1)), (0, True)) +
Piecewise((1, Eq(j, 1)), (0, True)))
assert ds(x*KD(i, k) + KD(j, k), (k, 2, 2)) == piecewise_fold(
Piecewise((x, Eq(i, 2)), (0, True)) +
Piecewise((1, Eq(j, 2)), (0, True)))
assert ds(x*KD(i, k) + KD(j, k), (k, 3, 3)) == piecewise_fold(
Piecewise((x, Eq(i, 3)), (0, True)) +
Piecewise((1, Eq(j, 3)), (0, True)))
assert ds(x*KD(i, k) + KD(j, k), (k, 1, l)) == piecewise_fold(
Piecewise((x, And(1 <= i, i <= l)), (0, True)) +
Piecewise((1, And(1 <= j, j <= l)), (0, True)))
assert ds(x*KD(i, k) + KD(j, k), (k, l, 3)) == piecewise_fold(
Piecewise((x, And(l <= i, i <= 3)), (0, True)) +
Piecewise((1, And(l <= j, j <= 3)), (0, True)))
assert ds(x*KD(i, k) + KD(j, k), (k, l, m)) == piecewise_fold(
Piecewise((x, And(l <= i, i <= m)), (0, True)) +
Piecewise((1, And(l <= j, j <= m)), (0, True)))
def test_deltasummation_mul_x_add_kd_kd():
assert ds(x*(KD(i, k) + KD(j, k)), (k, 1, 3)) == piecewise_fold(
Piecewise((x, And(1 <= i, i <= 3)), (0, True)) +
Piecewise((x, And(1 <= j, j <= 3)), (0, True)))
assert ds(x*(KD(i, k) + KD(j, k)), (k, 1, 1)) == piecewise_fold(
Piecewise((x, Eq(i, 1)), (0, True)) +
Piecewise((x, Eq(j, 1)), (0, True)))
assert ds(x*(KD(i, k) + KD(j, k)), (k, 2, 2)) == piecewise_fold(
Piecewise((x, Eq(i, 2)), (0, True)) +
Piecewise((x, Eq(j, 2)), (0, True)))
assert ds(x*(KD(i, k) + KD(j, k)), (k, 3, 3)) == piecewise_fold(
Piecewise((x, Eq(i, 3)), (0, True)) +
Piecewise((x, Eq(j, 3)), (0, True)))
assert ds(x*(KD(i, k) + KD(j, k)), (k, 1, l)) == piecewise_fold(
Piecewise((x, And(1 <= i, i <= l)), (0, True)) +
Piecewise((x, And(1 <= j, j <= l)), (0, True)))
assert ds(x*(KD(i, k) + KD(j, k)), (k, l, 3)) == piecewise_fold(
Piecewise((x, And(l <= i, i <= 3)), (0, True)) +
Piecewise((x, And(l <= j, j <= 3)), (0, True)))
assert ds(x*(KD(i, k) + KD(j, k)), (k, l, m)) == piecewise_fold(
Piecewise((x, And(l <= i, i <= m)), (0, True)) +
Piecewise((x, And(l <= j, j <= m)), (0, True)))
def test_deltasummation_mul_add_x_y_add_kd_kd():
assert ds((x + y)*(KD(i, k) + KD(j, k)), (k, 1, 3)) == piecewise_fold(
Piecewise((x + y, And(1 <= i, i <= 3)), (0, True)) +
Piecewise((x + y, And(1 <= j, j <= 3)), (0, True)))
assert ds((x + y)*(KD(i, k) + KD(j, k)), (k, 1, 1)) == piecewise_fold(
Piecewise((x + y, Eq(i, 1)), (0, True)) +
Piecewise((x + y, Eq(j, 1)), (0, True)))
assert ds((x + y)*(KD(i, k) + KD(j, k)), (k, 2, 2)) == piecewise_fold(
Piecewise((x + y, Eq(i, 2)), (0, True)) +
Piecewise((x + y, Eq(j, 2)), (0, True)))
assert ds((x + y)*(KD(i, k) + KD(j, k)), (k, 3, 3)) == piecewise_fold(
Piecewise((x + y, Eq(i, 3)), (0, True)) +
Piecewise((x + y, Eq(j, 3)), (0, True)))
assert ds((x + y)*(KD(i, k) + KD(j, k)), (k, 1, l)) == piecewise_fold(
Piecewise((x + y, And(1 <= i, i <= l)), (0, True)) +
Piecewise((x + y, And(1 <= j, j <= l)), (0, True)))
assert ds((x + y)*(KD(i, k) + KD(j, k)), (k, l, 3)) == piecewise_fold(
Piecewise((x + y, And(l <= i, i <= 3)), (0, True)) +
Piecewise((x + y, And(l <= j, j <= 3)), (0, True)))
assert ds((x + y)*(KD(i, k) + KD(j, k)), (k, l, m)) == piecewise_fold(
Piecewise((x + y, And(l <= i, i <= m)), (0, True)) +
Piecewise((x + y, And(l <= j, j <= m)), (0, True)))
def test_deltasummation_add_mul_x_y_mul_x_kd():
assert ds(x*y + x*KD(i, j), (j, 1, 3)) == \
Piecewise((3*x*y + x, And(1 <= i, i <= 3)), (3*x*y, True))
assert ds(x*y + x*KD(i, j), (j, 1, 1)) == \
Piecewise((x*y + x, Eq(i, 1)), (x*y, True))
assert ds(x*y + x*KD(i, j), (j, 2, 2)) == \
Piecewise((x*y + x, Eq(i, 2)), (x*y, True))
assert ds(x*y + x*KD(i, j), (j, 3, 3)) == \
Piecewise((x*y + x, Eq(i, 3)), (x*y, True))
assert ds(x*y + x*KD(i, j), (j, 1, k)) == \
Piecewise((k*x*y + x, And(1 <= i, i <= k)), (k*x*y, True))
assert ds(x*y + x*KD(i, j), (j, k, 3)) == \
Piecewise(((4 - k)*x*y + x, And(k <= i, i <= 3)), ((4 - k)*x*y, True))
assert ds(x*y + x*KD(i, j), (j, k, l)) == Piecewise(
((l - k + 1)*x*y + x, And(k <= i, i <= l)), ((l - k + 1)*x*y, True))
def test_deltasummation_mul_x_add_y_kd():
assert ds(x*(y + KD(i, j)), (j, 1, 3)) == \
Piecewise((3*x*y + x, And(1 <= i, i <= 3)), (3*x*y, True))
assert ds(x*(y + KD(i, j)), (j, 1, 1)) == \
Piecewise((x*y + x, Eq(i, 1)), (x*y, True))
assert ds(x*(y + KD(i, j)), (j, 2, 2)) == \
Piecewise((x*y + x, Eq(i, 2)), (x*y, True))
assert ds(x*(y + KD(i, j)), (j, 3, 3)) == \
Piecewise((x*y + x, Eq(i, 3)), (x*y, True))
assert ds(x*(y + KD(i, j)), (j, 1, k)) == \
Piecewise((k*x*y + x, And(1 <= i, i <= k)), (k*x*y, True))
assert ds(x*(y + KD(i, j)), (j, k, 3)) == \
Piecewise(((4 - k)*x*y + x, And(k <= i, i <= 3)), ((4 - k)*x*y, True))
assert ds(x*(y + KD(i, j)), (j, k, l)) == Piecewise(
((l - k + 1)*x*y + x, And(k <= i, i <= l)), ((l - k + 1)*x*y, True))
def test_deltasummation_mul_x_add_y_twokd():
assert ds(x*(y + 2*KD(i, j)), (j, 1, 3)) == \
Piecewise((3*x*y + 2*x, And(1 <= i, i <= 3)), (3*x*y, True))
assert ds(x*(y + 2*KD(i, j)), (j, 1, 1)) == \
Piecewise((x*y + 2*x, Eq(i, 1)), (x*y, True))
assert ds(x*(y + 2*KD(i, j)), (j, 2, 2)) == \
Piecewise((x*y + 2*x, Eq(i, 2)), (x*y, True))
assert ds(x*(y + 2*KD(i, j)), (j, 3, 3)) == \
Piecewise((x*y + 2*x, Eq(i, 3)), (x*y, True))
assert ds(x*(y + 2*KD(i, j)), (j, 1, k)) == \
Piecewise((k*x*y + 2*x, And(1 <= i, i <= k)), (k*x*y, True))
assert ds(x*(y + 2*KD(i, j)), (j, k, 3)) == Piecewise(
((4 - k)*x*y + 2*x, And(k <= i, i <= 3)), ((4 - k)*x*y, True))
assert ds(x*(y + 2*KD(i, j)), (j, k, l)) == Piecewise(
((l - k + 1)*x*y + 2*x, And(k <= i, i <= l)), ((l - k + 1)*x*y, True))
def test_deltasummation_mul_add_x_y_add_y_kd():
assert ds((x + y)*(y + KD(i, j)), (j, 1, 3)) == Piecewise(
(3*(x + y)*y + x + y, And(1 <= i, i <= 3)), (3*(x + y)*y, True))
assert ds((x + y)*(y + KD(i, j)), (j, 1, 1)) == \
Piecewise(((x + y)*y + x + y, Eq(i, 1)), ((x + y)*y, True))
assert ds((x + y)*(y + KD(i, j)), (j, 2, 2)) == \
Piecewise(((x + y)*y + x + y, Eq(i, 2)), ((x + y)*y, True))
assert ds((x + y)*(y + KD(i, j)), (j, 3, 3)) == \
Piecewise(((x + y)*y + x + y, Eq(i, 3)), ((x + y)*y, True))
assert ds((x + y)*(y + KD(i, j)), (j, 1, k)) == Piecewise(
(k*(x + y)*y + x + y, And(1 <= i, i <= k)), (k*(x + y)*y, True))
assert ds((x + y)*(y + KD(i, j)), (j, k, 3)) == Piecewise(
((4 - k)*(x + y)*y + x + y, And(k <= i, i <= 3)),
((4 - k)*(x + y)*y, True))
assert ds((x + y)*(y + KD(i, j)), (j, k, l)) == Piecewise(
((l - k + 1)*(x + y)*y + x + y, And(k <= i, i <= l)),
((l - k + 1)*(x + y)*y, True))
def test_deltasummation_mul_add_x_kd_add_y_kd():
assert ds((x + KD(i, k))*(y + KD(i, j)), (j, 1, 3)) == piecewise_fold(
Piecewise((KD(i, k) + x, And(1 <= i, i <= 3)), (0, True)) +
3*(KD(i, k) + x)*y)
assert ds((x + KD(i, k))*(y + KD(i, j)), (j, 1, 1)) == piecewise_fold(
Piecewise((KD(i, k) + x, Eq(i, 1)), (0, True)) +
(KD(i, k) + x)*y)
assert ds((x + KD(i, k))*(y + KD(i, j)), (j, 2, 2)) == piecewise_fold(
Piecewise((KD(i, k) + x, Eq(i, 2)), (0, True)) +
(KD(i, k) + x)*y)
assert ds((x + KD(i, k))*(y + KD(i, j)), (j, 3, 3)) == piecewise_fold(
Piecewise((KD(i, k) + x, Eq(i, 3)), (0, True)) +
(KD(i, k) + x)*y)
assert ds((x + KD(i, k))*(y + KD(i, j)), (j, 1, k)) == piecewise_fold(
Piecewise((KD(i, k) + x, And(1 <= i, i <= k)), (0, True)) +
k*(KD(i, k) + x)*y)
assert ds((x + KD(i, k))*(y + KD(i, j)), (j, k, 3)) == piecewise_fold(
Piecewise((KD(i, k) + x, And(k <= i, i <= 3)), (0, True)) +
(4 - k)*(KD(i, k) + x)*y)
assert ds((x + KD(i, k))*(y + KD(i, j)), (j, k, l)) == piecewise_fold(
Piecewise((KD(i, k) + x, And(k <= i, i <= l)), (0, True)) +
(l - k + 1)*(KD(i, k) + x)*y)
def test_extract_delta():
raises(ValueError, lambda: _extract_delta(KD(i, j) + KD(k, l), i))